The prior art contains several methods of phase ambiguity resolution for interferometric systems consisting of either collinear or non-collinear, coplanar arrangements of transducers, such as antennas. Referring to FIG. 1, in the case of an antenna array, which is sensitive to electromagnetic radiation, antenna elements A1 and A2 of the antenna array are presented with an electromagnetic wave emitted by a remote source. The wave is incident at the “phase centers” of each of the elements A1, A2 of the array from the exact same direction. This direction is referred to as either the direction of arrival (DOA) or the angle of arrival (AOA). Phase ambiguities arise under conditions in which the two antennas are further apart than one half wavelength of the signal carrier wave because practical phase comparators are incapable of discerning a phase angle outside of the range of ±π (±180°).
In a treatise published in 1973 by James E. Hanson titled “On Resolving Angle Ambiguities of n-Channel Interferometer Systems for Arbitrary Antenna Arrangements In a Plane” (Defense Technical Information Center Publication Number AD 776-335) addresses ambiguities. In this treatise, Hanson demonstrated how the problem of interferometric phase ambiguity resolution could be easily approached by casting the several differential phase measurements into direction cosine space as a set of equally spaced parallel straight lines; these straight lines arise from recasting the interferometer equation as a linear equation:
                                                        ψ              =                                                                                          2                      ⁢                                                                                          ⁢                      π                      ⁢                                                                                          ⁢                                              d                        y                                                              λ                                    ⁢                  sin                  ⁢                                                                          ⁢                  ϕ                  ⁢                                                                          ⁢                  sin                  ⁢                                                                          ⁢                  θ                                +                                                                            2                      ⁢                                                                                          ⁢                      π                      ⁢                                                                                          ⁢                                              d                        z                                                              λ                                    ⁢                                                                          ⁢                  cos                  ⁢                                                                          ⁢                  θ                                -                                  2                  ⁢                                                                          ⁢                  π                  ⁢                                                                          ⁢                  k                                                                                                                                                                  k                =                0                            ,                              ±                1                            ,                              ±                2                            ,              …              ⁢                                                          ,                                                          (        1        )            
wherein:    ψ is the measurable differential phase;    λ is the electromagnetic wavelength;    φ is the azimuth angle;    θ is the zenith angle;    k is an integer chosen to make ψ come out in the range of ±π; and    dy and dz are the y and z components of the inter-element baseline vector.
The meanings of the terms involved in equation (1) are illustrated in FIG. 1. In Hanson's representation sin φ sin θ and cos θ are replaced by Y and Z, respectively, and the new equation is manipulated so that it appears as:
                    Z        =                                            -                              (                                                      d                    y                                                        d                    z                                                  )                                      ⁢            Y                    +                                    λ                              2                ⁢                                                                  ⁢                π                ⁢                                                                  ⁢                                  d                  z                                                      ⁢                                          (                                  ψ                  +                                      2                    ⁢                                                                                  ⁢                    π                    ⁢                                                                                  ⁢                    k                                                  )                            .                                                          (        2        )            
Equation (2) is the equation of a set of parallel straight lines, one line for each value of the integer k. In addition, Hanson defines a unit circle as:(sin φ sin θ)2+(cos θ)2=1.  (3)
This unit circle describes the limits of visible space in that everywhere on and inside this unit circle (sin φ sin θ)2+(cos θ)2≦1. Accordingly, it is referred to as the unit circle of visibility. These sets of parallel lines along with the unit circle centered in direction cosine space are known to those familiar with Hanson's work as Hanson ambiguity diagram and the sets of straight lines are referred to as Hanson ambiguity trajectories. The entire set of trajectories completely describe the ambiguity performance of a linear or a non-linear, coplanar interferometer array (see FIGS. 2 and 2B).
According to Hanson, phase ambiguity resolution is accomplished by finding an arrangement of three or more antennas that create a Hanson ambiguity diagram with but a single point of intersection of the various trajectories, an intersection that is located in direction cosine space at the exact position of the radiating source; for strictly collinear arrays of antennas the single intersection is rather a single straight line. It is also noted that this single point of intersection in direction cosine space leads immediately to the two angles of arrival—φ the azimuth angle and θ the zenith angle—so that ambiguity resolution leads immediately to the determination of the angles of arrival (see FIG. 2A).
The differential phase measurements made with practical interferometers come with errors that arise due to systematic as well as thermodynamic perturbations within the array antennas and the receiving network. These errors cause the Hanson trajectories to move or shift randomly at right angles to the directions in which they lay. As a consequence, the single point of intersection in the ideal, no error condition becomes a set of pair-wise trajectory intersections (see FIG. 3). Thus, ambiguity resolution is accomplished by designing the ambiguity resolution computer algorithm so that it can discern a tightly grouped set of pair-wise intersections. Such an approach is described by Azzarelli, et al. in U.S. Pat. No. 6,140,963 but only for non-linear, coplanar arrays.
However, there is a need for a system and an ambiguity resolution method which can deal with non-coplanar arrangements of antenna elements. In addition, there is a need for a system and method which deal with the phase errors that arise due various perturbations and which deal with other than ideal conditions.